A multi-state approach and flexible payment distributions for microlevel reserving in general insurance

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1 A multi-state approach and flexible payment distributions for microlevel reserving in general insurance Katrien Antonio, Els Godecharle, and Robin Van Oirbeek AFI_16106 Electronic copy available at:

2 A multi-state approach and flexible payment distributions for micro-level reserving in general insurance Katrien Antonio 1, Els Godecharle *2, and Robin Van Oirbeek 3 1 Faculty of Economics and Business, KU Leuven, Belgium, and Faculty of Economics and Business, University of Amsterdam, The Netherlands. 2 Faculty of Economics and Business, KU Leuven, Belgium. 3 Faculty of Economics and Business, KU Leuven, Belgium, and Forsides Actuary, Belgium. April 9, 2016 Abstract Insurance companies hold reserves to be able to fulfill future liabilities with respect to the policies they write. Micro-level reserving methods focus on the development of individual claims over time, providing an alternative to the classical techniques that aggregate the development of claims into run-off triangles. This paper presents a discrete-time multi-state framework that reconstructs the claim development process as a series of transitions between a given set of states. The states in our setting represent the events that may happen over the lifetime of a claim, i.e. reporting, intermediate payments and closure. For each intermediate payment we model the payment distribution separately. To this end, we use a body-tail approach where the body of the distribution is modeled separately from the tail. Generalized Additive Models for Location, Scale and Shape introduced by Stasinopoulos and Rigby (2007) allow for flexible modeling of the body distribution while incorporating covariate information. We use the toolbox from Extreme Value Theory to determine the threshold separating the body from the tail and to model the tail of the payment distributions. We do not correct payments for inflation beforehand, but include relevant covariate information in the model. Using these building blocks, we outline a simulation procedure to evaluate the RBNS reserve. The method is applied to a real life data set, and we benchmark our results by means of a back test. Keywords: micro-level reserving, extreme value theory, splicing, multi-state model 1 Introduction An important feature in the insurance market is the precedence of premium income to the claim costs of an insurance policy. This characteristic is commonly referred to as an inverted production * Corresponding author: els.godecharle@gmail.com 1 Electronic copy available at:

3 1 Introduction 2 cycle. Due to this feature, it is important insurance companies hold sufficient reserves in order to be able to fulfill future liabilities with respect to claims that occur within the insurance coverage period. These reserves are a key factor on the liability side of the balance sheet of the insurance company. Accurate, reliable and stable reserving methods for a wide range of products and lines of business are crucial to safeguard solvency, stability and profitability. For example, according to Swiss Re (2008) deficient loss reserves were the main cause of financial insolvency in the US property and casualty (also called: general or non-life) market during the period With the introduction of new regulatory guidelines for the European insurance business in the form of Solvency II, the insurance industry has regained interest in using more elaborate methodology to model future cash flows and meet regulators increasing requirements. Insurance companies are strongly encouraged to replace their ad hoc, deterministic methods with fully stochastic approaches, aiming at accurately reflecting the riskiness in the portfolio under consideration. Current techniques for loss reserving will have to be improved, adjusted or extended to meet the requirements of the new regulations. This paper fits within this research avenue and contributes to the literature on statistical models for reserving in general insurance. Occurrence Development of one claim Payments Closure Run-off triangle Run-off time Reporting Payment All claims in portfolio Compress data Year of arrival s oc s 0 s 1 s 2 s 3 s 4 s c time Figure 1: Time line representing the development of a non-life claim and summarized in a run-off triangle. Figure 1 illustrates the run-off (or development) process of a non-life insurance claim (see Taylor (2000), England and Verrall (2002) and Wüthrich and Merz (2008)). A non-life insurance claim starts its lifetime or development at a certain point in time with the occurrence of an accident or claim event (e.g. a car accident). The occurrence date (s oc in Figure 1) of a claim refers to the date at which the claim event occurs. After the occurrence, the insured reports the claim to the insurance company. The reporting date (s 0 in Figure 1) refers to the date at which this happens. Once the insurer is aware of the claim and accepts the claim for reimbursement, some payments follow (at times s 1, s 2, s 3, s 4 in Figure 1) to compensate the insured for his loss. Eventually, when the loss covered by the policy is completely compensated for, the claim closes. The closing date (s c in Figure 1) refers to the date at which the claim closes. This finalizes the development of a claim. At the moment of evaluation (typically: end of a quarter, mid year or end of book year), say s, the insurer has to set reserves aside to fulfill his future liabilities and safeguard the solvency of the company. Loosely speaking, the insurer must predict, with maximum accuracy, the outstanding loss amount with respect to claims not yet closed at the moment of evaluation. Our interest goes out to the reserve necessary to cover outstanding liabilities from incurred claims that are not yet finalized. Existing methods for claims reserving (see England and Verrall (2002) and Wüthrich and Merz (2008)) are designed for aggregated data, conveniently summarized in a so-called run-off triangle. A run-off triangle summarizes the information registered on individual claims by aggregating

4 1 Introduction 3 payments into two-dimensional cells, representing the year of occurrence of the claim and the period of development during which the payment took place. See Figure 1 for a visualization of this process. Through this data compression many useful information of the claim is lost. In particular, policy(holder) characteristics (e.g. own risk or deductible, policy limit), characteristics of the accident, the claim, and expert information are ignored, or heavily compressed, in the run-off design. Recent literature challenges the appropriateness of reserving methods based on run-off triangles. For example, both Halliwel (2007) and Schiegl (2015) discuss and demonstrate conditions under which the traditional reserving methods for aggregated data, in particular the chain ladder method, are biased. A recent focus within the literature on loss reserving is the possible added value of using more extensive data when calculating reserves. Such data are available within insurance companies, as the left part of Figure 1 illustrates, and it fits within the avenue of big data and insurance analytics to explore their use, see Frees (2015). On the one hand, initiated by the work of Verrall et al. (2010), Martínez Miranda et al. (2012) and Martínez Miranda et al. (2013) extend the traditional chain ladder framework to Double Chain Ladder (DCL) and even continuous chain ladder setting. The DCL method combines the information of a classical run-off triangle with reported count data, whereas the continuous chain ladder improves the classical actuarial technique, which can be formulated as a histogram type of approach, by replacing this histogram by a kernel smoother. Hiabu et al. (2016a) and Hiabu et al. (2016b) show different ways to extend the traditional chain ladder framework through the inclusion of extra data resources in an aggregated format such as reporting delay and expert knowledge in the form of incurred payments. On the other hand, micro-level loss reserving models focus on the development of individual claims over time, as launched originally by Norberg (1993) and Norberg (1999). Haastrup and Arjas (1996) and Larsen (2007) model the development of a set of claims over time as a marked Poisson process. Antonio and Plat (2014) apply the continuous time framework of Norberg (1993) and Norberg (1999) to a real life data set, using maximum likelihood estimation, and add a discussion of distributional choices made in their analysis. In their approach, hazard rates drive the time to events in the development of a claim (e.g. a payment, or settlement) and a lognormal regression is used to model intermediate payments (e.g. those at time s 1, s 2, s 3 and s 4 in Figure 1) corrected for inflation using the Consumer Price Index (CPI). Zhao et al. (2009) and Zhao and Zhou (2010) also work in continuous time and propose a semi-parametric model to develop individual claims. Another strand within the micro-level reserving literature works in discrete time and aggregates payments per development period (e.g. a development year) while keeping focus on the development of an individual claim. Pigeon et al. (2014) extend the work of Pigeon et al. (2013), who introduce a claim specific run-off viewpoint with chain ladder like development factors, by including information on incurred losses. Drieskens et al. (2012) follow up on the work of Murphy and McLennan (2006) and propose to develop individual large claims through a non-parametric method based on historical simulation. Rosenlund (2012) presents a deterministic reserving method that allows to condition on specific characteristics of an individual claim. Godecharle and Antonio (2015) integrate this conditioning approach with the historical simulation used in Drieskens et al. (2012). Recent methodological work shows how the use of micro-level claims reserving methods significantly improves accuracy compared to the aggregate methods. Jin and Frees (2013) evaluate the performance of traditional runoff triangle techniques compared to a micro-level model, highlighting scenarios in which the latter outperform the former. Jin (2013) contribute to this topic in the form of a case study

5 2 A multi-state model for the development of a non life insurance claim 4 on a workers compensation insurance portfolio. Huang et al. (2015a), Huang et al. (2015b) and Huang et al. (2016) demonstrate, theoretically and numerically, the advantage one can gain from using a micro-level approach compared to the traditional run-off triangle techniques. We extend the micro-level loss reserving model of Antonio and Plat (2014) in multiple ways and connect it to recent contributions in the statistical literature on multiple state models, the framework of Generalized Additive Models for Location, Scale and Shape (GAMLSS) introduced by Stasinopoulos and Rigby (2007) and tools from Extreme Value Theory (EVT). As such, we avoid some of the rigid choices made in previous work on this topic. First, we adjust the loss reserving model of Antonio and Plat (2014) to a discrete-time framework. To this end, we propose a multi-state framework such that the claim development process can be reconstructed as a series of transitions between a given set of states. The use of a multi-state framework for pricing and reserving is common in a health insurance context (see Haberman and Pitacco (1999), Olivieri and Pitacco (2009), Chapter 6 in Pitacco (2014) and Czado and Rudolph (2002)) and in life insurance (see Chapter 8 in Dickson et al. (2013) and Chapter 20 in Frees et al. (2014)). Hesselager (1994) was one of the first to apply this popular actuarial model to the non-life insurance reserving context. A second extension lies in the modeling of the distribution of each subsequent payment in the development process. Contrary to Antonio and Plat (2014), we model the claim size distribution for each subsequent payment of the claim development process separately. Because the insurer is often confronted with heavy-tailed data and should safeguard the company against extreme losses, an accurate description of the upper tail of the payment distribution is of utmost importance. Therefore, as Larsen (2007) suggests, we model small and large payments separately and have specific attention for the tail of each of the payment distributions. Splicing (see Klugman et al. (2012) and Panjer (2006)) allows the use of a body-tail approach where the body of the distribution is modeled separately from the tail. GAMLSS introduced by Stasinopoulos and Rigby (2007) allow for flexible modeling of the the body distribution while incorporating covariate information in the location, scale and shape parameters for a wide collection of distributions. We use the toolbox from EVT (see McNeil et al. (2005) and Beirlant et al. (2006)) to determine the threshold separating the body from the tail and to model the tail of the payment distributions. Whereas our previous contributions, Antonio and Plat (2014), Pigeon et al. (2014) and Godecharle and Antonio (2015), corrected the observed payments beforehand for inflation based on the Consumer Price Index (CPI), we now do not apply any discounting and capture inflation effects directly by including appropriate covariates in our model. The paper is organized as follows. In Section 2 we describe the multi-state model for the development of a non-life insurance claim. Section 3 introduces the payment distribution model. We demonstrate our methodology in a case study on a data set from a European insurance company in Section 4 and end with a conclusion in Section 5. 2 A multi-state model for the development of a non life insurance claim 2.1 Claim dynamics Consider the non-life claim for which the development is illustrated in Figure 1. Insurance companies distinguish three types of claims depending on how far a claim is in the development

6 2 A multi-state model for the development of a non life insurance claim 5 process at the moment of evaluation s. For an IBNR or Incurred But Not Reported claim a claim event has happened, but the insurer is not aware of it yet at the moment of evaluation (i.e. s s oc and s < s 0 ). We call a claim RBNS or Reported But Not Settled when the insurer is aware of the claim, but the claim is not closed yet (i.e. s > s 0 and s s c ). Lastly, a claim is closed when we have observed its complete development at the moment of evaluation (i.e. s > s c ). The insurer has to set a reserve for both the IBNR and RBNS claims. In this paper, we present a framework to evaluate the RBNS reserve. We refer to Pigeon et al. (2013), Pigeon et al. (2014) and Antonio and Plat (2014) for a method to evaluate the IBNR reserve. 2.2 The multi-state approach We model the development of a non-life insurance claim as a sequence of events using the multistate model (S, T ) in Figure 2 with state space S and set of direct transitions T (Haberman and Pitacco (1999) and Denuit and Robert (2007)). At occurrence the claim starts its development in state S oc. Afterwards it is reported to the insurance company, corresponding to a transition to the reporting state S re := S 0. Once reported, a first payment can occur, implying a transition from state S 0 to state S 1. Just like S oc and S 0, the states S j (j {1,..., n p max 1}) are strictly transient, i.e. the claim can leave the state but not re-enter. Index j refers to the number of payments made in the past; thus a transition to state S j represents the j th payment of a claim. A claim in such a state can move on to another strictly transient state S j+1 or to one of the absorbing states, S tn or S tp, which are impossible to leave once entered. Index tn stands for Terminal, No payment, a transition to S tn means the claim closes without payment. The claim closes with a payment in case of a transition to S tp where tp stands for Terminal with Payment. The maximal number of payments throughout the development of a claim is denoted by n pmax. Therefore, the only possible transitions from state S npmax 1 are to the absorbing states (see Figure 2). A claim that moves from state S 0 directly to the absorbing state S tn does not receive any payments. The state space S is given by: S = {S oc, S re S 0, S 1, S 2,..., S np max 1, S tn, S tp }. (1) An event refers to the transition from one state in S to another. These events include claim occurrence, reporting, the j th payment (j {0, 1,..., n p max 1}) and closure with or without a payment. The set of direct transitions T defines all possible transitions in the multi-state model which are indicated by the arrows in Figure 2. We detect the three types of claims discussed in Section 2.1. A claim in state S oc is an IBNR claim because a claim event has happened, but the insurer is not aware of it as the claim did not make the transition to state S 0 yet. A claim in state S j (j {1,..., n p max 1}) is an RBNS claim. The insurer is aware of the claim, but the claim is not settled as it did not reach an absorbing state yet. Hence, the transition from S oc to S 0 terminates the IBNR part, but initializes the RBNS part in the development of a claim. A claim is closed when it reaches one of the absorbing states S tp or S tn. When a claim is in the RBNS part of the multi-state model in Figure 2 it faces competing risks (Klein and Moeschberger (2003), Pintilie (2006), Steele et al. (2006), Beyersmann et al. (2011) and Durrant et al. (2013)). A transition out of state S j implies no turning back. Moreover, the transition out of this state into S j+1, S tn or S tp are three competing events, as choosing one of the three for the latter transition excludes the possibility of choosing the other two events for the same transition.

7 2 A multi-state model for the development of a non life insurance claim 6 IBNR RBNS S oc λ oc,0 S re S 0 λ 0,tp λ 1,tp S tp λ 0,1 S 1 λ2,tp λ 1,2 λ npmax 1,tp S 2... S npmax 1 λ0,tn λ 1,tn λ 2,tn λ npmax 1,tn S tn Figure 2: Multi-state model for the development of a non-life insurance claim: states and hazard functions driving transitions between states. 2.3 A multinomial logit model for discrete-time transitions Time discretization As an insurer does not always record data from Figure 1 s timeline in continuous time, we approach the multi-state model in Figure 2 in discrete time. We use calender years in this paper, but the framework is easily extendable to other time discretizations. Denote the occurrence year of a claim κ by i(κ) {0, 1,..., n}. Hereby, n+1 is the number of occurrence years observed in the data set. We also record information on the development of the claim in discrete time and denote the development year by t {0, 1,... n} for which 0 corresponds to the occurrence year itself. To obtain a single payment per discrete time period, we aggregate all individual payments within the same time period (here: one calender year) into a single overall payment for that period. Define S(κ, t) S as the state occupied by claim κ at the end of development year t ( {0,..., n}). Then, S = {S(κ, t) t = 0, 1,..., n} is a discrete-time process describing the evolution of claim κ. As it is common for a claim to be reported and to receive its first payment within the same year, we allow the claim to transition from S oc to S 0 and from S 0 to S 1 in the same development year. For later transitions we consider a single transition per time period. We illustrate the time discretization and corresponding notation in Table 1 (see Section 4 for empirical illustrations using these data). Let us consider a data set where the observation period starts at 01/01/1997, making 1997 the first observed occurrence year. Therefore, claim κ originating in 1998 corresponds to occurrence year i(κ) = 1. Note that a double transition occurs in 1999: first the claim transitions from S oc to S 0 as the claim is reported, followed by a first payment in the same year implying a transition from S 0 to S 1. A transition from S 1 to S 2 happens in 2001 and corresponds to the second payment in the development of this claim for an amount of e400 (in total). Finally, a third payment follows in 2002 and the claim closes, which translates to a transition from S 2 to S tp in the multi-state framework. The transition into state S tp corresponds to the third payment. The other notation used in Table 1 is clarified throughout the remainder of the text.

8 2 A multi-state model for the development of a non life insurance claim 7 Event Date Development Payment Payment S(κ, t) Time to period t number event Continuous Discrete T Occurrence 08/23/ S oc Reporting 01/13/1999 S 0 T oc = 1 1 Cash flow e200 09/31/ Y 1 = 200 S 1 T 0 = 0 e150 03/15/2001 e250 10/20/2001 e100 12/03/2002 Closing 12/13/ Y 2 = 400 S 2 T 1 = Y 3 = 100 S tp T 2 = 1 Table 1: Illustration of the development of a claim κ as registered in continuous time (first three columns). Column 4 demonstrates the discrete time notation, columns 5 & 6 the payment number and corresponding payment amount, column 7 the stochastic process S(κ, t) and column 8 the development year T a at which the claim transitions out of a state S a given that time is reset upon arrival in state S a (Section 2.3 paragraph Multinomial logit model ). This claim is fictional. Multinomial logit model We use the hazard function λ a,b to model the transition from state S a to state S b in the multi-state model in Figure 2. Let T a denote the development year at which the claim moves out of state S a. We reset time at each transition. T a equal to zero implies the claim transitions into and out of state S a in the same year, which is the case for state S 0 in the illustration in Table 1. As T oc = T 2 = 1 in this example, the claim transitions out of respectively state S oc and S 2 in development year 1 since entering the latter state. The claim transitions out of state S 1 in development year 2 since entry in this state, so T 1 = 2. The label δ a tells us which of the competing risks the claim transitions to, where δ a = b means a transition to state S b. The hazard function from state S a to S b evaluated in τ, λ a,b (τ) = P (T a = τ, δ a = b T a τ), τ {0, 1,..., n} (2) is the probability the claim transitions from S a into S b in year τ, given that the transition out of state S a did not happen in the years before τ. We allow the hazard functions to depend on covariate information and use a multinomial logit framework to model the discrete-time hazard functions (Allison (1982) and Beyersmann et al. (2011), Chapter 7): exp{α a,b (τ) + β a,b λ a,b (τ x a,κ (τ)) = x a,κ(τ)} 1 + l exp{α a,l(τ) + β a,l x a,κ(τ)} where x a,κ (τ) is the (possibly) time-dependent covariate information for claim κ in year τ since transition into state S a, α a,b (τ) and β a,b are the regression parameters and the sum over l in the denominator runs over all possible states claim κ can directly transition into from state S a. For each transient state S a, the parameters used in the discrete-time hazard functions λ a,b (τ) are estimated simultaneously for all possible subsequent states S b using maximum likelihood 1. 1 We fit this model using the function multinom from the nnet R package introduced by Venables and Ripley (2002). (3)

9 3 A flexible body-tail model for the payment distribution 8 We hereby treat all observed time periods of the individual claims as separate, independent observations. Without additional covariate information x a,κ (τ), the specification in (3) simplifies to the Nelson- Aalen estimator (Nelson (1969), Aalen (1976)): ˆλ a,b (τ) = d a,b(τ) n a (τ) (4) with n a (τ) the number of claims in state S a at the start of the τ th year since transition into state S a and d a,b (τ) the number of these claims transitioning into state S b during year τ. 3 A flexible body-tail model for the payment distribution In the second part of our internal model, we model the payments in the development of a claim as visualized in Figure 1. We aggregate a claim s intermediate payments (at s 1,..., s 4 in Figure 1) within the same calendar year and obtain a single payment for that period. We label the payments consequently with a payment number, i.e. the first, second,... payment in the development of a claim. A separate model is built for each payment number. Let Y j (j {1,..., n pmax }) denote the j th payment of a random claim. For the example discussed in Section 2.3, Table 1 illustrates these payments in the fifth column. We need a model for Y j that is flexible, allows for the inclusion of claim-specific covariate information and correctly captures the skewness of the right tail. In the context of micro-level reserving Antonio and Plat (2014) include covariate information in the location and scale parameter of a lognormal distribution for intermediate payments which are not stratified by payment number. Distributional models for payments are also highly relevant in insurance pricing. Frees and Valdez (2008) model payments with a Generalized Beta of the second kind (GB2) distribution, with four parameters, including covariate information. Klein et al. (2014) use Generalized Additive Models for Location, Scale and Shape (GAMLSS) in a Bayesian framework for their observed claim severities, and specifically investigate the use of zero-adjusted versions of the gamma, inverse-gaussian and lognormal distributions where covariate information is included in three parameters (location and shape or scale parameter, as well as probability of a claim). Verbelen et al. (2015) propose mixtures of Erlangs as a flexible, yet tractable tool for loss modeling, while accounting for truncation and/or censoring. A model that fits the attritional losses (i.e. the small payments, also called the body of the loss distribution) does not necessarily capture large payments well. Pigeon and Denuit (2011) consider a composite lognormal-pareto model to capture both attritional and large losses. EVT (McNeil (1997), McNeil et al. (2005) and Beirlant et al. (2006)) suggests the use of the Generalized Pareto Distribution (GPD) to model the tail of the loss distribution, i.e. the losses above a certain, high threshold. As we want a flexible model for both attritional and large payments, we naturally opt for a global loss model, obtained as a spliced distribution with a body below the threshold and a tail above the threshold component. This way, we allow the density function of Y j to be a spliced distribution with two components (see Klugman et al. (2012), Panjer (2006) and Peters and Shevchenko (2015), Nadarajah and Bakar (2014), Aue and Kalkbrener (2006) for examples in modeling operational risk data): f Yj (y) = { pj,1 f j,1 (y), if 0 < y u j p j,2 f j,2 (y), if u j < y. (5)

10 3 A flexible body-tail model for the payment distribution 9 f j,1 (y) is a well-defined density function on the interval (0, u j ]. The tail of the distribution is treated separately by a well-defined density function f j,2 (y) on the interval (u j, ). We call u j the threshold which separates the body of the data from the tail, or the attritional losses from the large losses. The support of the body of the distribution is therefore (0, u j ] whereas the support of the tail is given by (u j, ). p j,1 is the probability (or weight) that Y j pertains to the body of the distribution whereas p j,2 expresses the probability of Y j belonging to the tail of the distribution, hence p j,1 + p j,2 = 1. We describe the choice of the splicing thresholds u j, components f j,1 and f j,2, probabilities p j,1 and p j,2 and the inclusion of claim specific covariate information in the remainder of this section. 3.1 Modeling the tail: threshold selection and GPD fit Let F j denote the cdf of Y j, the j th payment in the discretized development of a claim. We use the excess distribution over a threshold u to model the tail of F j : F ju (y) := P (Y j u y Y j > u) = F j(y + u) F j (u). (6) 1 F j (u) We determine the optimal threshold u j from which the tail of Y j starts by state of the art techniques from EVT (McNeil (1997), McNeil et al. (2005) and Beirlant et al. (2006)). The selected threshold, u j, is included in (5) as the threshold separating the body from the tail of the distribution. Balkema and de Haan (1974) and Pickands III (1975) prove that the excess distribution of common continuous distribution functions used in loss modeling converges to a Generalized Pareto Distribution (GPD) if the threshold u is high enough. Following their theorem, we assume F ju (y) = G(y) for some high threshold u where G is the GPD, defined as 1 ( 1 + γy ) 1 γ σ, y (0, ) if γ > 0 G(y) = 1 exp ( y ) σ, y (0, ) if γ = 0 1 ( (7) 1 + γy ) 1 γ σ, y (0, σ γ ) if γ < 0 where γ is the Extreme Value Index (EVI) and σ is a scale parameter. The selection procedure for the optimal threshold u j depends on the sign of the EVI. Graphical tools to determine this sign include the mean excess plot and the exponential QQ plot (see Section 4.1 in McNeil (1997), Section 7.2 in McNeil et al. (2005) and Section 1.2 in Beirlant et al. (2006)). For data with a GPD tail, the mean excess plot 2 becomes increasingly linear. The trend in the mean excess plot gives an indication of the sign of the EVI: a downward trend is associated with 2 The mean excess function of a random variable Y with finite mean, is defined as e(u) = E (Y u Y > u). (8) The mean excess function is empirically estimated as ny i=1 ê ny (u) = yi1 (u, )(y i) ny i=1 1 (u, )(y u (9) i) where y i is the i th observation and n Y the number of observations of Y. The mean excess plot is given by {Y ny k,n Y, ê ny (Y ny k,n Y ), 1 k n Y 1} where Y ny k,n Y denotes the (n Y k) th order statistic or the (k + 1) th largest observation of Y.

11 3 A flexible body-tail model for the payment distribution 10 γ < 0, a horizontal trend with γ = 0 and an upward trend with γ > 0 3. We illustrate this approach with the observations on the first payment, i.e. Y 1, from the case study presented in Section 4. Figure 3 shows the empirical mean excess plot (left) and the exponential QQ plot (right). The increasing empirical mean excess function indicates a heavytailed distribution (i.e. γ > 0), which is confirmed by the convex shape of the exponential QQ plot. We therefore conclude that the data used in this illustration are heavy-tailed (γ > 0). When modeling insurance losses (or: payments in a claim s run-off) we rarely observe the other cases in (7), i.e. γ = 0 or γ < 0. e^ny (Y ny k,ny ) Y ny k,ny Empirical quantiles Standard exponential quantiles Figure 3: Graphical tools to determine the sign of the EVI γ: sample mean excess plot (left) where the vertical line displays the selected threshold u j, and the exponential QQ plot (right). We illustrate the tools on first payment (Y 1 ) observations for the data set from Section 4 Threshold selection In the γ > 0 case, we determine the optimal threshold u j using the Hill (Section in McNeil et al. (2005) and Section 4.2 in Beirlant et al. (2006)), Zipf (Section 4.3 in Beirlant et al. (2006)) and second order Hill estimator (Section in Beirlant et al. (2006)) for the EVI γ in (7). We plot these estimators as a function of the number of tail observations taken into account, denoted by k, and look for a stable region in the graphs of these three γ- estimators. Figure 4 (top) visualizes the resulting estimates on the Y 1 data from our case study. We choose the threshold u 1 to be e15, (the 30 th largest observation of Y 1 ), indicated with the vertical line in the graph, at the point where the Hill and the second order Hill estimator meet. For other threshold selection methods we refer to Beirlant et al. (2006) and Scarrott and MacDonald (2012). Fitting the GPD tail Given the selected threshold u j, we use maximum likelihood estimation to determine the parameters in the GPD distribution for the exceedances above this threshold 4. Figure 5 shows the Peaks Over Threshold (POT) plot for the observations of Y 1 and the selected threshold. The horizontal line in this plot corresponds to the selected threshold u 1, whereas the height of each bar represents an observation on the first payment Y 1. For a payment above the selected threshold u 1, the length of the peak over the threshold represents the exceedance of the payment above u 1. The payments are grouped and color-coded per calendar year. We examine the goodness of fit of the GPD distribution using a QQ and PP plot as in Figure 4 (bottom). 3 The last observations in the mean excess plot average over a small number of excesses. As these values can distort the graph of the mean excess plot, we often do not consider these. 4 The gpd.fit function from the ismev package allows for this estimation in R.

12 3 A flexible body-tail model for the payment distribution 11 γ nd order Hill Hill Zipf k Empirical cdf cdf GPD Empirical quantiles GPD quantiles Figure 4: Graphical tools to determine the optimal threshold u j separating the body of the distribution from its tail: Hill, Zipf and second order Hill plot for the 200 largest observations (top) where the vertical line displays the selected threshold u j. The PP plot (bottom, left) and the QQ plot (bottom, right) of the GPD fit for exceedances above selected threshold u j. We illustrate the tools on the first payment (Y 1 ) observations for the data set from Section 4. These plots underline the good fit for the tail of the data observed on the first payment number, Y 1.

13 3 A flexible body-tail model for the payment distribution Payment Year Figure 5: The exceedances over the selected threshold u j as a Peaks Over Threshold (POT) plot: the length of each bar represents the height of an Y j observation, grouped and color-coded per calendar year. The horizontal line corresponds to the selected threshold u j. We illustrate the tool on the first payment (Y 1 ) observations for the data set from Section Modeling the body: a GAMLSS approach We use a parametric GAMLSS fit (Stasinopoulos and Rigby (2007)) in the body distribution, denoted by f j,1 (y) in (5). These flexible models extend the framework of Generalized Linear Models (GLM), widely used in actuarial tarification and reserving (De Jong and Heller (2008), Kaas et al. (2008) and Ohlsson and Johansson (2010)), by not restricting the distribution of the response variable to the exponential family. Moreover, they allow the inclusion of covariate information in up to four model parameters. Recent research demonstrates the usefulness of these models in a wide range of applications. For example to model mortgage loan losses as in Tong et al. (2013), to estimate measures of market risk as in Scandroglio et al. (2013), to analyze insurance data as in Klein et al. (2014) and income data as in Klein et al. (2015). We investigate the GAMLSS model specifications where the density function f j,1 (y x κ (t)) is conditional on at most four distribution parameters θ κ = (θ 1κ, θ 2κ, θ 3κ, θ 4κ ) = (µ κ, σ κ, ν κ, τ κ ). The first two parameters µ κ and σ κ in the density of claim κ are usually referred to as respectively location and scale parameter, whereas the remaining parameters, if present, are additional shape parameters. Notice that we drop the subscript j, 1 (see (5)) for these parameters to ease notation. We incorporate claim specific covariates in a linear way, namely g r (θ rκ ) = β rx κ where the parametric link function g r for r {1, 2, 3, 4} expresses the relation between the systematic component β rx κ and the distribution parameter θ rκ. Let x κ (t) be the vector with (possibly) time-dependent covariate information of claim κ in development year t. Given the covariate information, we model f j,1 (y x κ (t)) with a truncated parametric GAMLSS model since the observation on the j th payment in the development of a claim is left truncated by 0 and right-truncated by u j. We examine the goodness of fit of a

14 4 Case study 13 selection of distributions commonly used in loss modeling that are available within R s GAMLSS package 5. We use the AIC (see Akaike (1974)) to choose the preferred GAMLSS distribution for f j, Probability of belonging to the body or tail of a payment distribution The probabilities p j,r (x κ (t)) in (5) depend on covariate information x κ (t) of claim κ in development year t. We use a binomial logit framework to model these probabilities: p j,2 (x κ (t)) = exp{α + β x κ (t)} 1 + exp{α + β x κ (t)} p j,1 (x κ (t)) = 1 p j,2 (x κ (t)). (10) where α and β are the model parameters which we estimate by maximum likelihood 6. p j,1 (x κ (t)) and p j,2 represent the probability of belonging respectively to the body or tail of the distribution. 4 Case study We demonstrate the methodology on a data set from a European insurance company. The data consist of the bodily injury (BI) claims arising from general liability insurance contracts. Within the actuarial literature, Antonio and Plat (2014), Pigeon et al. (2013), Pigeon et al. (2014) and Godecharle and Antonio (2015) work with the same data set. These papers first discount payments using the relevant Consumer Price Index (CPI) and then model discounted payments. We choose not to correct the observed payments for inflation beforehand but instead want to capture inflation effects from the data at hand, using appropriate covariate information in our regression models. We split the data set in a training (January 1997 December 2004) and a validation data set (January 2005 August 2009) and we discretize time using annual periods starting from 01/01/1997 and running until 31/12/2004. As such we have eight full years of observations in the training data set. At the moment of evaluation, i.e. at the end of the day at 31/12/2004, the training data set contains 4,483 claims of which 3,452 are closed. The 1,031 remaining claims are RBNS claims for which we will simulate the reserve. One full simulation of the RBNS reserve consists of two steps. Starting from the multi-state model as depicted in Figure 2, we first use the estimated hazard functions from Section 2 to complete the stochastic process S(κ, t) for each RBNS claim κ by simulation. Define the maximum number of development years observed in the data set by n. In this application, when a claim reaches development year t = n = 8 and the claim is still open, we force a transition to closure without payment in the next development year such that S(κ, 9) = S tn, limiting the development of a claim in time. As a consequence n pmax = 9 in the multi-state model represented by Figure 2. Second, we use the calibrated distributions from Section 3 to simulate a payment for each simulated transition involving a payment. We incorporate the policy limit of e2.5 million from policy conditions underneath our data set. We construct the distribution for the RBNS reserve by repeating this two-step simulation process. 5 The gamlss.tr package in R allows for maximum likelihood parameter estimation of a GAMLSS distribution left-truncated by 0 and right-truncated by u j. 6 The nnet R package introduced by Venables and Ripley (2002) allows to compute the maximum likelihood estimators of these parameters.

15 4 Case study The multi-state model: estimation and simulation Summary statistics Table 2 shows summary statistics on the number of claims transitioning out of state S 0 (left) and state S 1 (right). Similar tables can be constructed for states S j (j {2,..., 8}). We recall the notation introduced in Section 2.3: n a (τ) is the number of claims in state S a at the start of the τ th year since transition into state S a and d a,b (τ) is the number of these claims transitioning into state S b during year τ. The top left cells in each table show the number n 0 (0) and n 1 (0) of claims we observe entering respectively state S 0 and S 1. Note that n 1 (0) consists of the claims that entered state S 0 (i.e. n 0 (0)) and did not transition into an absorbing state nor were censored before transitioning out of state S 0. Our model does not allow a claim to transition out of state S j in the year of entrance into this state when j 1. As a consequence, d 1,2 (0) = d 1,tp (0) = d 1,tn (0) = 0. We do allow a claim to transition out of state S 0 in the same year this state was entered, explaining the non-zero values for d 0,1 (0), d 0,tp (0), d 0,tn (0) in the left table. From the claims that entered state S 0, 1, 705 received a payment which did not close the claim (i.e. transition to S 1 ), 1,028 received a payment which closed the claim (i.e. transition to S tp ) and 309 claims closed without a payment (i.e. transition to S tn ) within the same year, τ = 0, of entrance into state S 0. The columns corresponding to τ = 1, 2 and 3 (left) and τ = 1, 2, 3 and 4 display these same numbers for claims that have been one, two and at least three years in state S 0, respectively state S 1. To make sure we have enough observations for each value of τ, we do not distinguish values of τ beyond 3 for S 0 and beyond 4 for S 0, but consider these as a single group. τ n 0 (τ) 4,483 1, d 0,1 (τ) 1, d 0,tp (τ) 1, d 0,tn (τ) τ n 1 (τ) 2,140 1, d 1,2 (τ) d 1,tp (τ) d 1,tn (τ) Table 2: Summary statistics on the transition out of S 0 (left) and out of S 1 (right).

16 4 Case study λ 0,1 λ 0,tp λ 0,tn λ 1,2 λ 1,tp λ 1,tn λ 0,1 λ 1,2 λ 0,tn λ 1,tn 0.75 λ 0,tp 0.75 λ 1,tp λ(τ) 0.50 λ(τ) >=3 τ >=4 τ Figure 6: Left, with τ the time since entry in state S 0 : ˆλ 0,1 (τ) (full black line), ˆλ 0,tn (τ) (dotted green line), ˆλ 0,tp (τ) (dashed blue line) and the probability of not making a transition 1 ˆλ 0,1 (τ) ˆλ 0,tn (τ) ˆλ 0,tp (τ) (dotted-dashed red line). Right, as a function of time since entry τ in state S 1 : ˆλ1,2 (τ) (full black line), ˆλ 1,tn (τ) (dotted green line), ˆλ 1,tp (τ) (dashed blue line) and the probability of not making a transition 1 ˆλ 1,2 (τ) ˆλ 1,tn (τ) ˆλ 1,tp (τ) (dotted-dashed red line). Estimation results We use the multinomial logit model from (3) without additional covariate information x a,κ (τ), resulting in the Nelson-Aalen estimator (4), to estimate the hazard functions λ a,b (τ). We leave the inclusion of covariate information as a subject for future research. To illustrate our results, Figure 6 visualizes the estimated hazard functions for transitions out of state S 0 and S 1. Simulation of future paths for RBNS claims The stochastic process S(κ, t) describes the movement of a claim κ through the multi-state model as time evolves. For an RBNS claim, this process did not yet reach an absorbing state at the moment of evaluation. We simulate the further run-off of such an RBNS claim by simulating its future path driven by estimated hazard functions ˆλ a,b until S(κ, t) reaches an absorbing state or until t reaches the maximum number n of observed development years. The simulation procedure for claim κ starts with determining the last state this claim occupies in its development process S(κ, t), say S a, and the first unobserved development period τ since entrance into this state. We then select the relevant hazard function values ˆλ a,b (τ) for all possible transitions to some state b out of state S a in development period τ. Given these probabilities we simulate what happens in period τ:

17 2e 02 1e 02 0e+00 4e 02 2e 02 0e e 02 1e 02 0e+00 8e 02 6e 02 4e 02 2e 02 0e e 02 2e 02 1e 02 0e+00 9e 02 6e 02 3e 02 0e e 02 3e 02 2e 02 1e 02 0e+00 9e 03 6e 03 3e 03 0e Total Figure 7: Distribution for the number of payments from the RBNS claims per calendar year obtained by the simulation procedure (2,000 simulations) based on the estimated hazard rates. The point labeled is used to indicate number of payments observed in the validation data set of RBNS claims. Note that the validation data set contains observations until August As a consequence, the number of payments observed in 2009 is incomplete and we do not observe payments in 2010 and The total number of payments is therefore also incomplete. 4 Case study 16

18 4 Case study 17 a non-terminal payment transition (to state S a+1 ), a terminal payment transition (S tp ), closure without payment (S tn ) or no transition at all (S a ). In case the simulated state is an absorbing one, the stochastic process S(κ, t) ends here as the claim closes. If the development period t reaches the maximum number n = 8 of observed development years without the claim attaining an absorbing state, we close the claim by a transition to state S tn in the following period. When the claim did not reach an absorbing state and t < n, we repeat the procedure and simulate the next state in the stochastic process of the open claim. For each RBNS claim observed in the training data set, we simulate its future path through the multi-state model. By repeating this procedure, we can construct empirical distributions of quantities of interest using the simulated paths. For example, Figure 7 shows the empirical distribution of the number of payments per calendar year (i.e. 2005, 2006, and so on). Table 3 shows summary statistics of these empirical distributions per calendar year. We simulate 2,000 paths and compare the simulated number of payments to the actual observed number of payments in the validation data set. Note that the year 2009 is only observed until August in the validation data set, making the observed number for this calendar year incomplete. Calendar year Total Minimum ,129 25% quantile ,220 Median ,245 Mean ,245 75% quantile ,269 95% quantile ,307 Maximum ,388 erved NA NA 1, 085 Table 3: Summary statistics for the number of payments from the RBNS claims per calendar year obtained by the simulation procedure (2,000 simulations) based on the estimated hazards. The row labeled erved shows the number of payments observed in the validation data set of RBNS claims for the corresponding calendar year. Figure 7 and Table 3 show the empirical distribution captures the number of payments observed in the validation data set in calendar year 2005 very well. For calendar year 2006, 2007 and 2008 the observed value lies more in the left tail of the distribution. Results may probably be improved by introducing covariate information in the hazard functions, as suggested in Section 5. The observed number of payments in 2009 and in total are also in the left tail of the empirical distribution. Note, however, that we only observe 2009 until August in the validation data set, making these observed numbers incomplete. As a consequence, we do not observe any payments in 2010 and 2011 and the total observed number is incomplete as well. 4.2 Flexible payment distributions: estimation and simulation Descriptive statistics Table 4 summarizes the empirical distribution per payment number as observed in the training data set. Recall from Section 3 that we stratify the distribution of non-zero payments in a claim s run-off based on payment number and use notation Y j for the j th payment during the development of a claim. We construct the spliced density function of Y j (see (5)) per payment number j, following the procedure outlined in Section 3. Because we have very few payment observations for high

19 4 Case study 18 payment numbers, in particular: payment numbers 5 8 in this data set, we fit one body-tail model on the collection of these observations. We denote the payment distribution of these payment numbers by f Y5 and the corresponding random variable by Y 5. Payment Y 1 Y 2 Y 3 Y 4 Y 5 observed 3,655 1, Minimum Mean 1, , , , , Median , , , , % quantile 5, , , , , Maximum 162, , , , , Table 4: Training data set, with occurrence years : summary statistics of the empirical distribution per payment number. Tail of the distribution We apply the strategy from Section 3.1 to all payment number distributions. Graphics illustrating this strategy were shown in Section 3.1. We apply the same strategy to Y 2, Y 3, Y 4 and Y 5. In all cases we observe an increasing mean excess plot and convexity of the exponential QQ plots (cfr. our illustration in Figure 3 with data observed on Y 1 ). Thus, we conclude that Y 1, Y 2, Y 3, Y 4 and Y 5 are heavy-tailed. Table 5 shows the selected thresholds. For example, the threshold splicing the distribution of Y 2 is u 2 = e12, which corresponds to the 83 rd largest observation. The GPD fit on the tail of Y 1,..., Y 5 is verified by means of a QQ and a PP plot. Table 6 shows the corresponding parameter estimates and their standard error (in brackets). Note that the standard errors for the highest two payment numbers (Y 4 and Y 5 ) becomes large as a result of the low number of observations included in the fit. u 1 u 2 u 3 u 4 u 5 Threshold e15, e12, e19,129.6 e28, e29, Order statistic Table 5: Optimal splicing thresholds u j for j {1, 2, 3, 4, 5} together with their corresponding order statistic. f 1,2 f 2,2 f 3,2 f 4,2 f 5,2 ˆγ 0.46 (0.27) 0.51 (0.17) 0.51 (0.26) 0.54 (0.36) 0.49 (0.74) ˆσ 10, (3,258.57) 7, (1,420.20) 12, (3,229.18) 24, (7,844.73) 25, (28,345.51) Table 6: Parameter estimates and their standard error (in brackets) for f,2 in equation (5) with the GPD distribution given by (7).